If $A^{2} + 3 I = 0$ find $A^{- 1}$ .

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Solution

Given that,
$A^{2} + 3 I = 0$

By multiplying on both sides with $A^{- 1}$ we get,

$A^{2} . A^{- 1} + 3 I . A^{- 1} = 0. A^{- 1}$

$A . (A . A^{- 1}) + 3 (I . A^{- 1}) = 0$

$A . I + 3 A^{- 1} = 0$

$A + 3 A^{- 1} = 0$

$3 A^{- 1} = 0 - A = - A$

$∴ A^{- 1} = \frac{1}{3} (- A)$

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